Robin van Persie has waited a long time to score at Liverpool and will remember both of the goals that earned Arsenal the points here. His first is the only goal he has scored with his head this season; his second was a winner in the second minute of added time that will rank as one of the best strikes of his career, even though he makes a habit of unanswerable finishes. This volley was not quite in the Marco van Basten class of spectacular, nor was it as jaw-droppingly sumptuous as the one Van Persie scored against Everton this season, but it was a gloriously instinctive piece of skill that silenced Anfield just when it seemed a ninth home draw of the season was on the cards.

It was almost a carbon copy of his goal against Everton, even down to Alex Song providing the pass, though given the stage for the game and the significance of the victory in the race for fourth place in the Premier League, this was a cause for greater celebration.

Having gone ahead through an own goal and missed another penalty, Liverpool were probably only worth a point, but when Van Persie reached on to Song's long ball forward in stoppage time they were denied even that. There could be no argument against such a decisive score and there was little time left in any case. Van Persie hit the ball as it dropped over his shoulder and surprised Pepe Reina with his speed of thought rather than the power or placement of his shot.

"When you see the ball landing at Van Persie's feet you can always expect something special," Arsène Wenger said. "With a striker of that quality in the side you always have a chance until you are killed off. Our goalkeeper and centre forward made the difference today because we were not in the game in the first half. One-one at the interval flattered us."

Liverpool should have gone in front after 17 minutes of an eventful first half when Luis Suárez won a penalty, tumbling over Wojciech Szczesny's outstretched leg after pushing the ball past the goalkeeper. There was some doubt about the amount of contact – and at least a suggestion that Suárez was looking to go down – but it appeared that a foul had been committed so the referee, Mark Halsey, pointed to the spot. Perhaps significantly, he took no further action against the Arsenal goalkeeper, who could have been dismissed but stayed on the field to save his side twice over from Dirk Kuyt's penalty.

First Szczesny dived to his right to keep out the spot-kick, then flung himself to his left to palm away the follow-up, which Kuyt might have been better to leave to the better-placed Jordan Henderson. "The second was the save of the game," said Arsène Wenger. "I wasn't happy about the referee's decision originally, but I heard television had shown it was a penalty."

Szczesny's good work was wasted six minutes later, when he was beaten by one of his own defenders. Liverpool had put together a good passing move, with Jay Spearing and Stewart Downing launching Henderson down the right wing. The eventual cross was nothing special and, from a good position to intercept, Laurent Koscielny came up with a finish better than any of the home strikers would have managed, smacking the ball past his keeper under no particular pressure.

Following up after Szczesny had pushed out a shot from Henderson, Suárez hit a post as Liverpool enjoyed their best spell, but just as Arsenal seemed to be struggling to contain their opponents they drew level, with a goal of clinical quality produced out of nothing.

Bacary Sagna was allowed all the time he required to arrow in a pinpoint cross from the right that took Martin Skrtel out of the equation and left Van Persie with the relatively straightforward task of staying in front of Jamie Carragher and heading the ball past Reina from the six-yard line for what was his 30th goal of the season.

Suárez was unlucky towards the end of the first half when he turned Thomas Vermaelen inside out only to see Szczesny just reach his shot, then Liverpool hit the woodwork again just before the interval when Kuyt touched on Charlie Adam's cross. Liverpool have been denied by the goal-frame more often than any other Premier League club this season and while some will say that shows the extent of their attacking intent, others would suggest there is room for improvement in their finishing.

Arsenal lost Mikel Arteta with a neck injury at the start of the second half after an accidental collision with Henderson, but continued to hold their own against a Liverpool midfield that lacked Steven Gerrard, injured while playing for England in midweek.

They might have taken the lead on 70 minutes when Theo Walcott's shot took a late deflection off Skrtel to bring a sharp reaction save from Reina, though they could also have gone behind at around the same stage in the game. Kuyt's excellent cross from the right fell to Martin Kelly, at the far post, who finished like a full-back suffering a nosebleed from being too high up the pitch.

To his credit, Kenny Dalglish did not attempt to claim an injustice had taken place or that Liverpool were only beaten by a wonder strike. "The scoreline didn't reflect the game, but that was our own fault," the Liverpool manager said. "We played well and did enough to have won, but we were beaten because we didn't score enough goals when we were dominating the game. The result was the only disappointing thing. The next lesson we need to learn might be how to be ugly and win."